Four-dimensional Haken cobordism theory

TL;DR

This paper establishes the existence of 4-dimensional Haken cobordisms connecting any two closed Haken 3-manifolds, demonstrating that each such 3-manifold can be realized as a boundary component of a Haken 4-manifold.

Contribution

It proves the existence of 4-dimensional Haken cobordisms with prescribed boundary components, advancing the understanding of Haken manifold cobordism theory.

Findings

Existence of 4D Haken cobordisms between any two closed Haken 3-manifolds

Each closed Haken 3-manifold can be realized as a boundary of a Haken 4-manifold

Injectivity of fundamental group inclusion in Haken cobordisms

Abstract

Cobordism of Haken $n$-manifolds is defined by a Haken $(n+1)$-manifold $W$ whose boundary has two components, each of which is a closed Haken $n$-manifold. In addition, the inclusion map of the fundamental group of each boundary component to $\pi_1(W)$ is injective. In this paper we prove that there are 4-dimensional Haken cobordisms whose boundary consists of any two closed Haken 3-manifolds. In particular, each closed Haken 3-manifold is the $\pi_1$-injective boundary of some Haken 4-manifold.